Asymptotic normality and valid inference for Gaussian variational approximation

TL;DR

This paper establishes the asymptotic normality of Gaussian variational estimators in a Poisson mixed model, enabling valid statistical inference and demonstrating good coverage properties through simulations.

Contribution

It provides the first detailed asymptotic analysis of Gaussian variational estimators, confirming their statistical validity and practical effectiveness.

Findings

Gaussian variational estimators are asymptotically normal.

Confidence intervals based on these estimators have good coverage.

Variational approximation achieves similar precision to exact likelihood methods.

Abstract

We derive the precise asymptotic distributional behavior of Gaussian variational approximate estimators of the parameters in a single-predictor Poisson mixed model. These results are the deepest yet obtained concerning the statistical properties of a variational approximation method. Moreover, they give rise to asymptotically valid statistical inference. A simulation study demonstrates that Gaussian variational approximate confidence intervals possess good to excellent coverage properties, and have a similar precision to their exact likelihood counterparts.